Understanding training and generalization in deep learning by Fourier analysis. - Zhiqin John Xu (CoRR, 2018)

Maximum Entropy Principle Analysis in Network Systems with Short-time Recordings. - Zhi-Qin John Xu; Jennifer Crodelle; Douglas Zhou; David Cai (CoRR, 2018)

Frequency Principle in Deep Learning with General Loss Functions and Its Potential Application. - Zhi-Qin John Xu (CoRR, 2018)

Training Behavior of Deep Neural Network in Frequency Domain. - Zhi-Qin John Xu; Yaoyu Zhang; Yanyang Xiao (ICONIP (1), 2019)

Frequency Principle: Fourier Analysis Sheds Light on Deep Neural Networks. - Zhi-Qin John Xu; Yaoyu Zhang; Tao Luo 012; Yanyang Xiao; Zheng Ma 009 (CoRR, 2019)

Explicitizing an Implicit Bias of the Frequency Principle in Two-layer Neural Networks. - Yaoyu Zhang; Zhi-Qin John Xu; Tao Luo 012; Zheng Ma 009 (CoRR, 2019)

Theory of the Frequency Principle for General Deep Neural Networks. - Tao Luo 012; Zheng Ma 009; Zhi-Qin John Xu; Yaoyu Zhang (CoRR, 2019)

Multi-scale Deep Neural Networks for Solving High Dimensional PDEs. - Wei Cai 003; Zhi-Qin John Xu (CoRR, 2019)

Dynamical and Coupling Structure of Pulse-Coupled Networks in Maximum Entropy Analysis. - Zhi-Qin John Xu; Douglas Zhou; David Cai (Entropy, 2019)

A type of generalization error induced by initialization in deep neural networks. - Yaoyu Zhang; Zhi-Qin John Xu; Tao Luo 012; Zheng Ma 009 (MSML, 2020)

Implicit bias with Ritz-Galerkin method in understanding deep learning for solving PDEs. - Jihong Wang; Zhi-Qin John Xu; Jiwei Zhang 002; Yaoyu Zhang (CoRR, 2020)

Multi-scale Deep Neural Network (MscaleDNN) for Solving Poisson-Boltzmann Equation in Complex Domains. - Ziqi Liu; Wei Cai 003; Zhi-Qin John Xu (CoRR, 2020)

A regularized deep matrix factorized model of matrix completion for image restoration. - Zhemin Li; Zhi-Qin John Xu; Tao Luo 012; Hongxia Wang (CoRR, 2020)

Fourier-domain Variational Formulation and Its Well-posedness for Supervised Learning. - Tao Luo 012; Zheng Ma 009; Zhiwei Wang; Zhi-Qin John Xu; Yaoyu Zhang (CoRR, 2020)

Deep Frequency Principle Towards Understanding Why Deeper Learning Is Faster. - Zhiqin John Xu; Hanxu Zhou (AAAI, 2021)

Embedding Principle of Loss Landscape of Deep Neural Networks. - Yaoyu Zhang; Zhongwang Zhang; Tao Luo 012; Zhi-Qin John Xu (NeurIPS, 2021)

Frequency Principle in Deep Learning Beyond Gradient-descent-based Training. - Yuheng Ma 002; Zhi-Qin John Xu; Jiwei Zhang 002 (CoRR, 2021)

Linear Frequency Principle Model to Understand the Absence of Overfitting in Neural Networks. - Yaoyu Zhang; Tao Luo 012; Zheng Ma 009; Zhi-Qin John Xu (CoRR, 2021)

An Upper Limit of Decaying Rate with Respect to Frequency in Deep Neural Network. - Tao Luo 012; Zheng Ma 009; Zhiwei Wang; Zhi-Qin John Xu; Yaoyu Zhang (CoRR, 2021)

Towards Understanding the Condensation of Two-layer Neural Networks at Initial Training. - Zhi-Qin John Xu; Hanxu Zhou; Tao Luo 012; Yaoyu Zhang (CoRR, 2021)

MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEs. - Lulu Zhang; Tao Luo 012; Yaoyu Zhang; Zhi-Qin John Xu; Zheng Ma 009 (CoRR, 2021)

Force-in-domain GAN inversion. - Guangjie Leng; Yekun Zhu; Zhi-Qin John Xu (CoRR, 2021)

Data-informed Deep Optimization. - Lulu Zhang; Zhi-Qin John Xu; Yaoyu Zhang (CoRR, 2021)

A variance principle explains why dropout finds flatter minima. - Zhongwang Zhang; Hanxu Zhou; Zhi-Qin John Xu (CoRR, 2021)

Embedding Principle: a hierarchical structure of loss landscape of deep neural networks. - Yaoyu Zhang; Yuqing Li; Zhongwang Zhang; Tao Luo 012; Zhi-Qin John Xu (CoRR, 2021)

Phase Diagram for Two-layer ReLU Neural Networks at Infinite-width Limit. - Tao Luo 012; Zhi-Qin John Xu; Zheng Ma 009; Yaoyu Zhang (J. Mach. Learn. Res., 2021)

An Upper Limit of Decaying Rate with Respect to Frequency in Linear Frequency Principle Model. - Tao Luo 012; Zheng Ma 010; Zhiwei Wang; Zhiqin John Xu; Yaoyu Zhang (MSML, 2022)

Empirical Phase Diagram for Three-layer Neural Networks with Infinite Width. - Hanxu Zhou; Qixuan Zhou; Zhenyuan Jin; Tao Luo 012; Yaoyu Zhang; Zhi-Qin John Xu (NeurIPS, 2022)

Towards Understanding the Condensation of Neural Networks at Initial Training. - Hanxu Zhou; Qixuan Zhou; Tao Luo 012; Yaoyu Zhang; Zhi-Qin John Xu (NeurIPS, 2022)

A deep learning-based model reduction (DeePMR) method for simplifying chemical kinetics. - Zhiwei Wang; Yaoyu Zhang; Yiguang Ju; Weinan E; Zhi-Qin John Xu; Tianhan Zhang (CoRR, 2022)

A multi-scale sampling method for accurate and robust deep neural network to predict combustion chemical kinetics. - Tianhan Zhang; Yuxiao Yi; Yifan Xu; Zhi X. Chen; Yaoyu Zhang; Weinan E; Zhi-Qin John Xu (CoRR, 2022)

Overview frequency principle/spectral bias in deep learning. - Zhi-Qin John Xu; Yaoyu Zhang; Tao Luo 012 (CoRR, 2022)

An Experimental Comparison Between Temporal Difference and Residual Gradient with Neural Network Approximation. - Shuyu Yin; Tao Luo 012; Peilin Liu; Zhi-Qin John Xu (CoRR, 2022)

Embedding Principle in Depth for the Loss Landscape Analysis of Deep Neural Networks. - Zhiwei Bai; Tao Luo 012; Zhi-Qin John Xu; Yaoyu Zhang (CoRR, 2022)

Linear Stability Hypothesis and Rank Stratification for Nonlinear Models. - Yaoyu Zhang; Zhongwang Zhang; Leyang Zhang; Zhiwei Bai; Tao Luo 012; Zhi-Qin John Xu (CoRR, 2022)

A regularised deep matrix factorised model of matrix completion for image restoration. - Zhemin Li; Zhi-Qin John Xu; Tao Luo 012; Hongxia Wang (IET Image Process., 2022)

On the Exact Computation of Linear Frequency Principle Dynamics and Its Generalization. - Tao Luo 012; Zheng Ma 009; Zhi-Qin John Xu; Yaoyu Zhang (SIAM J. Math. Data Sci., 2022)

Phase Diagram of Initial Condensation for Two-layer Neural Networks. - Zhengan Chen; Yuqing Li; Tao Luo 012; Zhangchen Zhou; Zhi-Qin John Xu (CoRR, 2023)

Laplace-fPINNs: Laplace-based fractional physics-informed neural networks for solving forward and inverse problems of subdiffusion. - Xiong-Bin Yan; Zhi-Qin John Xu; Zheng Ma 010 (CoRR, 2023)

Understanding the Initial Condensation of Convolutional Neural Networks. - Zhangchen Zhou; Hanxu Zhou; Yuqing Li; Zhi-Qin John Xu (CoRR, 2023)

Loss Spike in Training Neural Networks. - Zhongwang Zhang; Zhi-Qin John Xu (CoRR, 2023)

Solving a class of multi-scale elliptic PDEs by means of Fourier-based mixed physics informed neural networks. - Xi'an Li 004; Jinran Wu; Zhi-Qin John Xu; You-Gan Wang (CoRR, 2023)

Optimistic Estimate Uncovers the Potential of Nonlinear Models. - Yaoyu Zhang; Zhongwang Zhang; Leyang Zhang; Zhiwei Bai; Tao Luo 012; Zhi-Qin John Xu (CoRR, 2023)

An Unsupervised Deep Learning Approach for the Wave Equation Inverse Problem. - Xiong-Bin Yan; Keke Wu; Zhi-Qin John Xu; Zheng Ma 010 (CoRR, 2023)

Limitation of Characterizing Implicit Regularization by Data-independent Functions. - Leyang Zhang; Zhi-Qin John Xu; Tao Luo 012; Yaoyu Zhang (Trans. Mach. Learn. Res., 2023)

Subspace decomposition based DNN algorithm for elliptic type multi-scale PDEs. - Xi-An Li 004; Zhi-Qin John Xu; Lei Zhang (J. Comput. Phys., 2023)

DeepFlame: A deep learning empowered open-source platform for reacting flow simulations. - Runze Mao; Minqi Lin; Yan Zhang; Tianhan Zhang; Zhi-Qin John Xu; Zhi X. Chen (Comput. Phys. Commun., 2023)

Stochastic Modified Equations and Dynamics of Dropout Algorithm. - Zhongwang Zhang; Yuqing Li; Tao Luo 012; Zhi-Qin John Xu (ICLR, 2024)

Initialization is Critical to Whether Transformers Fit Composite Functions by Reasoning or Memorizing. - Zhongwang Zhang; Pengxiao Lin; Zhiwei Wang; Yaoyu Zhang; Zhi-Qin John Xu (NeurIPS, 2024)

Solving multiscale dynamical systems by deep learning. - Zhi-Qin John Xu; Junjie Yao; Yuxiao Yi; Liangkai Hang; Weinan E; Yaoyu Zhang; Tianhan Zhang (CoRR, 2024)

Anchor function: a type of benchmark functions for studying language models. - Zhongwang Zhang; Zhiwei Wang; Junjie Yao; Zhangchen Zhou; Xiaolong Li; Weinan E; Zhi-Qin John Xu (CoRR, 2024)

Understanding Time Series Anomaly State Detection through One-Class Classification. - Hanxu Zhou; Yuan Zhang; Guangjie Leng; Ruofan Wang; Zhi-Qin John Xu (CoRR, 2024)

Loss Jump During Loss Switch in Solving PDEs with Neural Networks. - Zhiwei Wang; Lulu Zhang; Zhongwang Zhang; Zhi-Qin John Xu (CoRR, 2024)

Initialization is Critical to Whether Transformers Fit Composite Functions by Inference or Memorizing. - Zhongwang Zhang; Pengxiao Lin; Zhiwei Wang; Yaoyu Zhang; Zhi-Qin John Xu (CoRR, 2024)

Towards Understanding How Transformer Perform Multi-step Reasoning with Matching Operation. - Zhiwei Wang; Yunji Wang; Zhongwang Zhang; Zhangchen Zhou; Hui Jin; Tianyang Hu; Jiacheng Sun; Zhenguo Li; Yaoyu Zhang; Zhi-Qin John Xu (CoRR, 2024)

A rationale from frequency perspective for grokking in training neural network. - Zhangchen Zhou; Yaoyu Zhang; Zhi-Qin John Xu (CoRR, 2024)

Implicit Regularization of Dropout. - Zhongwang Zhang; Zhi-Qin John Xu (IEEE Trans. Pattern Anal. Mach. Intell., 2024)

Efficient and Flexible Method for Reducing Moderate-Size Deep Neural Networks with Condensation. - Tianyi Chen; Zhi-Qin John Xu (Entropy, 2024)

Solving multiscale dynamical systems by deep learning. - Junjie Yao; Yuxiao Yi; Liangkai Hang; Weinan E; Weizong Wang; Yaoyu Zhang; Tianhan Zhang; Zhi-Qin John Xu (Comput. Phys. Commun., 2025)

Bayesian Inversion with Neural Operator (BINO) for modeling subdiffusion: Forward and inverse problems. - Xiong-Bin Yan; Zhi-Qin John Xu; Zheng Ma 010 (J. Comput. Appl. Math., 2025)

Input gradient annealing neural network for solving Fokker-Planck equations with low temperature. - Liangkai Hang; Dan Hu; Zhi-Qin John Xu (J. Comput. Phys., 2025)

On understanding and overcoming spectral biases of deep neural network learning methods for solving PDEs. - Zhi-Qin John Xu; Lulu Zhang; Wei Cai 003 (J. Comput. Phys., 2025)

Complexity Control Facilitates Reasoning-Based Compositional Generalization in Transformers. - Zhongwang Zhang; Pengxiao Lin; Zhiwei Wang; Yaoyu Zhang; Zhi-Qin John Xu (CoRR, 2025)

Reasoning Bias of Next Token Prediction Training. - Pengxiao Lin; Zhongwang Zhang; Zhi-Qin John Xu (CoRR, 2025)

An Analysis for Reasoning Bias of Language Models with Small Initialization. - Junjie Yao; Zhongwang Zhang; Zhi-Qin John Xu (CoRR, 2025)

An overview of condensation phenomenon in deep learning. - Zhi-Qin John Xu; Yaoyu Zhang; Zhangchen Zhou (CoRR, 2025)

MemOS: An Operating System for Memory-Augmented Generation (MAG) in Large Language Models. - Zhiyu Li; Shichao Song; Hanyu Wang; Simin Niu; Ding Chen; Jiawei Yang; Chenyang Xi; Huayi Lai; Jihao Zhao; Yezhaohui Wang; Junpeng Ren; Zehao Lin; et al. (CoRR, 2025)

Scalable Complexity Control Facilitates Reasoning Ability of LLMs. - Liangkai Hang; Junjie Yao; Zhiwei Bai; Tianyi Chen; Yang Chen; Rongjie Diao; Hezhou Li; Pengxiao Lin; Zhiwei Wang; Cheng Xu; Zhongwang Zhang; Zhangchen Zhou; et al. (CoRR, 2025)

Adaptive Preconditioners Trigger Loss Spikes in Adam. - Zhiwei Bai; Zhangchen Zhou; Jiajie Zhao; Xiaolong Li; Zhiyu Li; Feiyu Xiong; Hongkang Yang; Yaoyu Zhang; Zhi-Qin John Xu (CoRR, 2025)

MemOS: A Memory OS for AI System. - Zhiyu Li; Shichao Song; Chenyang Xi; Hanyu Wang; Chen Tang; Simin Niu; Ding Chen; Jiawei Yang; Chunyu Li; Qingchen Yu; Jihao Zhao; Yezhaohui Wang; et al. (CoRR, 2025)